Gauge transformation and reciprocal link for (2+1)-dimensional integrable field systems

It is well known that approach of the classical R-matrix formalism to the specific infinitedimensional Lie algebras can be used for systematic construction of field and lattice integrable dispersive systems (soliton systems) as well as dispersionless integrable field systems (see [1]-[17] and the references there). The Lie algebra of pseudo-differential operators (PDO) leads to the construction of (1+1)-dimensional integrable soliton systems [15, 8, 4]. Considering the (1+1)-dimensional integrable hierarchies with infinitely many fields one can extract from them closed equations for a single field by the elimination of the remaining fields [14, 7, 11]. This method, the so-called Sato approach [14], leads to construction of (2+1)-dimensional integrable one-field equations: Kadomtsev-Petviashvili (KP), modified Kadomtsev-Petviashvili (mKP) and (2+1) Harry-Dym (HD). An analogous method with matrix coefficients and dressing operators, the so-called matrix Sato theory [8], permits a construction of (2+1)-dimensional integrable evolution equations, with more number of fields, like (2+1) AKNS. There is another more effective and systematic method for construction of (2+1)-dimensional integrable systems, the so-called central extension procedure [16, 17, 13]. The central extension approach to integrable field, lattice-field and dispersionless systems was presented in [6, 5] and [3]. In this paper an appropriate restrictions of Lax operators coming from centrally extended PDO algebra, are systematically considered. In [9, 11] a wide class of gauge, reciprocal, Bäcklund and auto-Bäcklund transformations for (1+1)-dimensional soliton systems and (2+1)-dimensional systems like KP, mKP and HD, originating from the PDO Lie algebra, is presented. Therefore, the investigation of such transformations for (2+1)-dimensional systems, originating from the centrally extended PDO algebra, seems interesting. In this paper the relations between three classes of Lax hierarchies, coming from the centrally extended PDO algebra, are constructed.